
theorem Th36:
  for G1, G2 being _Graph, F being continuous PGraphMapping of G1, G2
  st rng F_V = the_Vertices_of G2 holds G1 is loopless implies G2 is loopless
proof
  let G1, G2 be _Graph, F be continuous PGraphMapping of G1, G2;
  assume A1: rng F_V = the_Vertices_of G2;
  assume A2: G1 is loopless;
  for v being object holds not ex e being object st e Joins v,v,G2
  proof
    let v be object;
    given e9 being object such that
      A3: e9 Joins v,v,G2;
    v in rng F_V by A1, A3, GLIB_000:13;
    then consider v0 being object such that
      A4: v0 in dom F_V & F_V.v0 = v by FUNCT_1:def 3;
    consider e being object such that
      A5: e Joins v0,v0,G1 & e in dom F_E & F_E.e = e9 by A3, A4, Def16;
    thus contradiction by A2, A5, GLIB_000:18;
  end;
  hence G2 is loopless by GLIB_000:18;
end;
